#154996

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂⁴	C₂³×C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,3,4)(5,6)(7,8)(9,10,11,12)(15,19)(17,20)(22,25)(23,27);(2,4)(5,6,7,8)(22,25)(26,28);(5,6)(7,8)(13,14,16,18)(15,17,19,20)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,6)(7,8)(13,15)(14,17)(16,19)(18,20)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=efgh,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62192);